26-05-2012, 03:58 PM

Superposition Modulation: Myths and Facts

[23198]

INTRODUCTION

Since Shannon’s landmark contribution in 1948,

it has been known that any communication channel

is characterized by a so-called channel capacity.

Channel capacity is, in simple words, the

maximum achievable throughput in bits per

channel symbol (called transmission rate) subject

to the constraint of quasi-error-free transmission.

Operating near capacity implies power efficiency

and simultaneously bandwidth efficiency. A

transmission scheme is called power efficient if

the average transmit power per information bit

for a given quality constraint is sufficiently small.

Discussions on “green radio,” for example, indicate

that power efficiency is becoming even

more important nowadays. A transmission

scheme is called bandwidth efficient if many

information bits can be transmitted per time unit

in a given bandwidth. Particularly in wireless

communications, we experience that bandwidth

is sparse and becoming really expensive.

THE PRINCIPLE OF SUPERPOSITION MODULATION

In order to explain the principle of superposition

modulation, let us consider the summation of

two random sequences with elements +1 and –1,

respectively. For example, the first sequence is

[+1, –1, +1, –1], the second sequence [+1, –1,

–1, +1]. Each sequence is referred to as a layer.

Trivially, the sum is [+2, –2, 0, 0] if both

sequences are equally weighted by one. Given a

received value of ±2, the mapping can uniquely

be reversed. However, if the received value is 0,

detection is ambiguous, because it is unclear

whether +1 –1 or –1 + 1 has been superimposed.

As shown later, SM with real-valued equal

weighting is always non-bijective. In our example

N = 2 bits are mapped onto just M = 3 signal

points, where M < 2N.

PROPERTIES OF SUPERPOSITION MODULATION

Given these preliminary remarks, we are now

ready to formally define SM using complex baseband

notation [6, 10]. As shown in Fig. 1, a binary

data stream is first split into N parallel binary

data streams. Let bn ∈ {0, 1} denote the nth

information bit for a certain time slot, 1 ≤ n ≤ N.

All bits bn are mapped onto binary antipodal

symbols dn ∈ {+1, –1}. Afterward, the symbols

dn are weighted by a set of complex-valued factors

αn ∈CI in order to obtain N chips cn in parallel.

This weighting corresponds to power and

phase allocation and is crucial for the performance

of SM.

EQUAL POWER ALLOCATION

Considering power normalization, in the case of

equal power allocation (EPA) the weighting factors

are fixed according to αn = 1/√

—

N for all 1 ≤

n ≤ N. It can easily be proven that the number of

distinct symbols x (i.e., the cardinality of the

symbol alphabet X) is equal to ⎪X⎪ = N + 1.

Since N + 1 < 2N, SM-EPA is always non-bijective.

An interesting interpretation is that SMEPA

incorporates a lossy source encoder, a fact

that is explored later. The data symbols are

binomial distributed. For N → ∞, the binomial

distribution approaches a Gaussian distribution.

GROUPED POWER ALLOCATION

So far, we observed that SM-EPA is non-bijective.

Due to near-Gaussian quadrature components,

no active shaping is necessary for

achieving the capacity. The main bottleneck,

however, is that H(X) ~ log2(N). In contrast,

SM-UPA is bijective, but not capacity achieving

and hence not further elaborated.

[23198]

INTRODUCTION

Since Shannon’s landmark contribution in 1948,

it has been known that any communication channel

is characterized by a so-called channel capacity.

Channel capacity is, in simple words, the

maximum achievable throughput in bits per

channel symbol (called transmission rate) subject

to the constraint of quasi-error-free transmission.

Operating near capacity implies power efficiency

and simultaneously bandwidth efficiency. A

transmission scheme is called power efficient if

the average transmit power per information bit

for a given quality constraint is sufficiently small.

Discussions on “green radio,” for example, indicate

that power efficiency is becoming even

more important nowadays. A transmission

scheme is called bandwidth efficient if many

information bits can be transmitted per time unit

in a given bandwidth. Particularly in wireless

communications, we experience that bandwidth

is sparse and becoming really expensive.

THE PRINCIPLE OF SUPERPOSITION MODULATION

In order to explain the principle of superposition

modulation, let us consider the summation of

two random sequences with elements +1 and –1,

respectively. For example, the first sequence is

[+1, –1, +1, –1], the second sequence [+1, –1,

–1, +1]. Each sequence is referred to as a layer.

Trivially, the sum is [+2, –2, 0, 0] if both

sequences are equally weighted by one. Given a

received value of ±2, the mapping can uniquely

be reversed. However, if the received value is 0,

detection is ambiguous, because it is unclear

whether +1 –1 or –1 + 1 has been superimposed.

As shown later, SM with real-valued equal

weighting is always non-bijective. In our example

N = 2 bits are mapped onto just M = 3 signal

points, where M < 2N.

PROPERTIES OF SUPERPOSITION MODULATION

Given these preliminary remarks, we are now

ready to formally define SM using complex baseband

notation [6, 10]. As shown in Fig. 1, a binary

data stream is first split into N parallel binary

data streams. Let bn ∈ {0, 1} denote the nth

information bit for a certain time slot, 1 ≤ n ≤ N.

All bits bn are mapped onto binary antipodal

symbols dn ∈ {+1, –1}. Afterward, the symbols

dn are weighted by a set of complex-valued factors

αn ∈CI in order to obtain N chips cn in parallel.

This weighting corresponds to power and

phase allocation and is crucial for the performance

of SM.

EQUAL POWER ALLOCATION

Considering power normalization, in the case of

equal power allocation (EPA) the weighting factors

are fixed according to αn = 1/√

—

N for all 1 ≤

n ≤ N. It can easily be proven that the number of

distinct symbols x (i.e., the cardinality of the

symbol alphabet X) is equal to ⎪X⎪ = N + 1.

Since N + 1 < 2N, SM-EPA is always non-bijective.

An interesting interpretation is that SMEPA

incorporates a lossy source encoder, a fact

that is explored later. The data symbols are

binomial distributed. For N → ∞, the binomial

distribution approaches a Gaussian distribution.

GROUPED POWER ALLOCATION

So far, we observed that SM-EPA is non-bijective.

Due to near-Gaussian quadrature components,

no active shaping is necessary for

achieving the capacity. The main bottleneck,

however, is that H(X) ~ log2(N). In contrast,

SM-UPA is bijective, but not capacity achieving

and hence not further elaborated.